Properties of long quantum walks in one and two dimensions
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The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena by performing a detailed numerical study of discrete-time quantum walks. In one dimension (1D), we compute the structure of the probability distribution, which is not a smooth curve but shows oscillatory features on all length scales. By analyzing walks up to N = 1,000,000 steps, we discuss the scaling characteristics and limiting forms of the QW in both real and Fourier space. In 2D, with a view to ready experimental realization, we consider two types of QW, one based on a four-faced coin and the other on sequential flipping of a single two-faced coin. Both QWs may be generated using two two-faced coins, which in the first case are completely unentangled and in the second are maximally entangled. We draw on our 1D results to characterize the properties of both walks, demonstrating maximal speed-up and emerging semi-classical behavior in the maximally entangled QW.
KeywordsDiscrete-time quantum walk Structure of probability distribution Scaling characteristics Pseudo-binomial distribution on the edge of two dimensional quantum walk
The authors gratefully acknowledge helpful discussions with Professor B. Normand. Work at Renmin University was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11174365 and by the National Basic Research Program of China (NBRPC) under Grant No. 2012CB921704. PX was supported by the NSFC under Grant Nos. 11174052 and 11474049, by the NBRPC under Grant No. 2011CB921203, by the Open Fund from the State Key Laboratory of Precision Spectroscopy of East China Normal University, and by the CAST Innovation Fund.
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